Porisma

A hint to get started

Use different regions for area and revolution

The graph shows y=1/(1+x2)y=1/(1+x^2) and y=tan⁡xy=\tan x, where the source prints −π/2≤x≤π/2-\pi/2\le x\le\pi/2.

(i) Find the area bounded by the two curves and x=1x=1.

(ii) Find the volume generated when the region bounded by the two curves and the y-axis is rotated by 2π2\pi radians about the y-axis.

CJC · Lecture · Example 19

The Teacher shades and labels distinct regions (i) and (ii), adds x=1, intersection guides and a y-axis rotation arrow, and supplies both integrals and the inverse-boundary algebra. These annotations are integrated into the steps and reconstructed figure.

The boundary controlling the disc radius changes at y=0.71981; split the y integral there. Inverse tangent uses radians.

Read the worked solution